Critical currents and vortex-unbinding transitions in quench-condensed ultrathin films
of bismuth and tin
K. Das Gupta, Swati S. Soman, G. Sambandamurthy,* and N. Chandrasekhar†
Department of Physics, Indian Institute of Science, Bangalore, India
We have investigated the I-V characteristics of strongly disordered ultrathin films of Bi and Sn produced by
quench condensation. Our results show that both these systems can be visualized as strongly disordered arrays
of Josephson junctions. The experimentally observed I-V characteristics of these films are hysteretic when the
injected current is ramped from zero to critical current and back. These are remarkably similar to the hysteretic
I-V of an underdamped single junction. We show by computer simulations that hysteresis can persist in a very
strongly disordered array. It is also possible to estimate the individual junction parameters (R, C, and I c ) from
the experimental I-V of the film using this model. The films studied are in a regime where the Josephsoncoupling energy is larger than the charging energy. We find that a simple relation I c (T)⫽I c (0) 关 1⫺(T/T c ) 4 兴
describes the temperature dependence of the critical current quite accurately for films with sheet resistance
⬃500 ⍀ or lower. We also find evidence of a vortex-unbinding transition in the I-V taken at temperatures
slightly below the mean-field T c .
I. INTRODUCTION
Ultrathin films produced by quench condensation under
highly reproducible conditions, with extensive atomic-scale
disorder, have been investigated over the past two decades as
model systems to study the interplay between disorder, interactions, and superconductivity. More than a dozen single and
multicomponent systems—Al, Bi, Be, Ga, In, Pb, Sn, Nb,
Mo-C, Mo-Ge, etc.1–9,11—have been shown to exhibit an
insulator-superconductor transition as their thickness is increased or disorder is reduced. The sheet resistance at which
the transition takes place has been observed to vary from
⬇3 k⍀ in Mo-C films9 to ⬇20 k⍀ in Al on Ge.10 The nature of this transition in the T→0 limit has attracted maximum attention,12,13 with the parameters characterizing the
superconducting state having received comparatively lesser
attention. It is also unclear at this point what the order parameter of this transition is and what similarities in physics
the phenomenon has with metal-insulator transitions in twodimensional electron gas 共2DEG兲 systems. Most of the reported data on these systems do not show hysteresis in I-V
characteristics, the reason being that in most cases they are
not probed with currents close to the critical currents of these
films. Many physically relevant microstructural parameters
can, however, be estimated from the hysteretic I-V.
In this paper we present some experimental results on
quench-condensed superconducting films of Bi and Sn. The
choice of the two materials was dictated by certain experimental considerations, to be discussed below. Earlier studies
had classified materials like Sn, Pb, and Ga as ‘‘granular’’
and Bi 共particularly Bi quenched on a thin Ge underlayer兲 as
‘‘homogeneous.’’ The basis of this classification lay in the
rather different resistive transitions to the superconducting
state exhibited by these films. In Pb, Ga, and Sn a metallic
phase appears to be sandwiched between the insulating and
superconducting phases. This observation has stimulated the
study of a Bose-metal phase in the T→0 limit.12 It was first
suggested by Ramakrishnan14 that a ‘‘phase-only’’ picture
was appropriate for the granular films. Destruction of superconductivity in these films was thought to take place by destruction of phase coherence 共via Josephson coupling兲 between
superconducting
grains
embedded
in
a
nonsuperconducting matrix. In the homogeneous films 共Bi
has been taken to be a representative of this class兲, on the
other hand, destruction of superconductivity was thought to
take place via destruction of the amplitude of the wave function. Though the resistive transitions (R-T traces兲 as a function of increasing 共decreasing兲 thickness 共disorder兲 for Bi
and Sn show qualitative differences, we show using experimentally obtained I-V characteristics and computer simulations that both these systems can be understood as disordered
arrays of Josephson junctions.
Some recent scanning tunneling microscopy 共STM兲
studies15,16 on these films suggest that the structure of the
first monolayer of atoms that stick to the substrate may be
rather different than the subsequent upper layers. Cluster
sizes in the range of ⬃30 Å were observed in films nominally 10 monolayers thick. We discuss the relevance of these
to the observed properties of our films. The structure of these
films remains a mystery. Recent STM data are an indication
of what one may expect, but since they were taken under
conditions different from experiments such as those of Henning et al.,17 Goldman and co-workers,1,5,6 and this work,
comparisons of microstructure should be made with an element of caution.
To summarize, there are significant unresolved problems
regarding the processes that can lead to formation of such
clusters at very low (⬍10 K) substrate temperatures. Landau
and co-workers18 –20 pointed out that the formation of structures such as those reported by Ekinci and Valles15,16 require
that incoming atoms be able to perform ⬃500 hops before
they finally settle to their positions. This order of diffusivity
is much more than what incoming atoms striking cold substrates are believed to have, although such a possibility can-
not be ruled out. We have not addressed this question, but
show that several properties of the superconducting state in
these films may be understood if we accept the presence of
superconducting and nonsuperconducting regions in close
proximity in the film, a case of microstructure similar to that
reported by Ekinci and Valles.15,16 Our model bears a close
resemblance to the ‘‘percolation-type’’ model used by Meir21
in describing the M -I transition in 2DEG and has been discussed in our earlier work.3
II. EXPERIMENTAL SETUP
The experiments were done in a UHV cryostat, custom
designed for in situ study of ultrathin films, described in our
earlier publications.2,3 A turbomolecular pump, backed by an
oil-free diaphragm pump, provided a vacuum better than
10⫺8 Torr. The substrate was either a-quartz or crystalline
sapphire, mounted with adequate thermal contact on a copper
cold finger, in contact with a pumped helium bath. The lowest attainable temperature was 1.8 K. The material 共e.g., Bi,
Sn兲 was evaporated from a pyrolytic boron nitride crucible in
a Knudsen cell 共effusion cell兲 of the type used in molecular
beam epitaxy 共MBE兲. The temperature of the cell was carefully controlled to give a steady deposition rate between 1
and 5 Å/min. The required temperatures were 650 °C for Bi
and 1150 °C for Sn.
Two Hall-bar type masks defined the sample; the actual
sample size was 6 mm⫻1 mm. One of the samples had a
predeposited Ge underlayer of ⬇10 Å thickness. This geometry enables us to evaluate the claim that Ge underlayers may
significantly improve the wetting properties of the films and
the possibility that there may be screening effects due to the
presence of an underlayer of a dielectric constant larger than
that of the substrate. The metal flux reached the sample
through carefully aligned holes in successive cryoshields
cooled by liquid helium and nitrogen. These shields also reduce the heat load on the sample and provide cryopumping,
for better vacuum in the neighborhood of the sample. Electrical contacts to the films were provided through predeposited platinum contact pads, about 50 Å thick. Four-probe dc
measurements were carried out using a standard highimpedance current source and a nanovoltmeter or an electrometer.
FIG. 1. Resistance-temperature (R-T) curves of a set of quenchcondensed Sn films showing the insulator-superconductor transition.
where R Q ⫽h/4e 2 . With the approximations stated above, for
a film with R 䊐 ⬇500 ⍀ and ␤ ⬇2, we get E J /E c ⬇70. This
places the films for which we find a simple empirical relation
between the critical current and temperature, well into the
regime E J ⰇE c .
The I-V characteristics of a single Josephson junction
共JJ兲, in the framework of the RCSJ model, is well known and
is discussed in several texts.22 If the transfer of a Cooper pair
across the junction causes negligible change in the phase
difference ( ␾ ) across the junction, then ␾ can be viewed as
a continuous variable and the I-V characteristics 共neglecting
the effect of thermal noise兲 can be deduced from Eq. 共2兲:
␤␾¨ ⫹ ␾˙ ⫹sin ␾ ⫽i.
共2兲
˙ 典 gives the observed voltage drop.
The long-time average 具 ␾
i is the normalized current, I/I c , through the junction. ␤ is
III. OBSERVATIONS AND DISCUSSION
The R-T traces of a series of Sn films exhibiting the I-S
transition are shown in Fig. 1. Figure 2 shows a typical set of
observed I-V characteristics for a Bi film in the superconducting regime. We can read off the critical and retrapping
currents for the whole film. We claim here 共and demonstrate
later兲 that this ratio would be nearly same as that of a single
junction. For a single junction we have the exact relation
between the Josephson coupling energy, charging energy,
normal-state resistance, and the Stewart-McCumber junction
parameter ␤ :
E J /E c ⫽ 共 2/␲ 2 兲共 R Q /R N 兲 2 ␤ ,
共1兲
FIG. 2. Current-voltage (I-V) characteristics of Bi films on sapphire. The observed characteristics are similar for Bi and Sn on
crystalline as well as amorphous substrates.
FIG. 3. Schematic of a 5⫻5 JJ array. Each bond is a parallel
combination of three elements, as shown. The current is injected
and extracted as shown; in the other direction we use periodic
boundary condition.
related to the microscopic parameters of the junction as ␤
⫽(2eI c R N /ប)R N C and also fixes the ratio of the retrapping
to critical currents (I r /I c ). The larger the value of ␤ , smaller
is this ratio and wider the hysteresis.
We first investigate, by computer simulations, whether the
hysteresis would persist in a 2D RCSJ array even if all the
junction parameters are allowed to have a large distribution
of values 共i.e., in presence of strong disorder兲. Also we need
to know whether the observed I-V curve would be significantly altered by the particular type of distribution 共e.g.,
square, Gaussian, log-normal兲. We address these questions in
the section on computer simulation, the details of which are
presented later.
An important fact about a 2D network of resistances is the
following: not too close to the percolation threshold, the observed resistance, measured between two edges 共see Fig. 3兲,
is close to the average value of all the resistances. In a 1D
chain it is obviously the sum of all the resistances, in a 3D
lattice the measured resistance is much lower than the average. These can be verified by simple numerical calculations.
In 2D even when the width of the distribution is more than
90% of the mean value, the measured normal-state sheet resistance (R 䊐 ) of the film does not differ from the mean of all
the resistances by more than 10%. We have verified by direct
numerical calculation that this result holds good for array
sizes down to a 10⫻10 array. If there are N junctions in
parallel, then the observed critical current of the array would
be approximately N times the average critical current of one
junction. This allows us to infer some important facts. This
also demonstrates our claim above that the ratio of the critical current to the retrapping current for an array is the same
as that for a single junction in the array. In what follows, this
will be further substantiated.
A. Hysteresis in a disordered array
If identical junctions were laid out on an array, the I-V
characteristics would be indistinguishable from that of a
single junction in absence of a magnetic field. However,
FIG. 4. Computer-simulated I-V curves of a 16⫻16 array of
Josephson junctions, on a square lattice. The bond parameters
(R,C,I c ) are chosen from a square distribution—‘‘w’’ denotes the
width of the distribution as a fraction of the mean value of the
parameters. The mean values are used to calculate ␤ . Use of a
‘‘log-normal’’ distribution also leads to similar results.
when there is a large spread in the values of the junction
parameters, it is not a priori obvious what their behavior will
be. This is because different junctions with different critical
currents may undergo transitions at different currents, and for
a large array the sharp transition might be broken into many
steps and then rounded off by finite-temperature effects. Our
main result in this regard is that, even in a small array (16
⫻16), disorder does not completely destroy the hysteresis—
until about 50% disorder 共ratio of width to mean value in a
square distribution兲 the shape of the I-V loop does not
change appreciably. We have shown the results 共see Fig. 4兲,
using a square distribution of disorder here, though simulations using a log-normal distribution 共results not shown兲 also
lead to similar conclusion.
1. Driving equations and the algorithm
In an array the algebraic sum of all the currents meeting at
any node must be zero—this is required by Kirchoff’s current law. As shown in Fig. 3, the current is fed uniformly
through one edge of the array and extracted through the opposite. In the direction perpendicular to that of current injection, we generally use periodic boundary conditions; this is
equivalent to joining the remaining two free edges of the
array together. From the current conservation equations we
get a total of N 2 coupled second-order differential equations.
The model we are considering neglects the self-inductance of
the array. Consider a single node 共not on any of the edges兲, at
which the phase at some instant is ␾ . In the four neighboring
nodes the phases are ␾ u , ␾ l , ␾ b , ␾ r —where the subscripts
denote up, left, bottom, and right, respectively. Every bond
will have its characteristic R, C, and I c value—since the
array is disordered. We use R 0 , C 0 , and I C0 to denote the
average values. With a little algebra we can show that the
contribution of any link to a node is
¨ ⫹r ␾
˙ ⫹i sin ␾ ,
i⫽c ␤ 0 ␾
where, for a completely ordered array r⫽1, c⫽1 for all
bonds. Adding all the current contributions to a node we get
␤ 0 关共 c l ⫹c u ⫹c r ⫹c b 兲 ṗ⫺c l ˙
p l ⫺c u ˙
p u ⫺c r ˙
p r ⫺c b ˙
pb兴
⫹ 共 1/r l ⫹1/r u ⫹1/r r ⫹1/r b 兲 p⫺ 共 p l /r l ⫹p u /r u
⫹p r /r r ⫹p b /r b 兲 ⫹ 共 i l s l ⫹i u s u ⫹i r s r ⫹i b s b 兲 ⫽0,
2. ‘‘Fast’’ algorithm for regular arrays
共3兲
˙ l,u,r,b ⫽p l,u,r,b and sin(␾⫺␾l,u,r,b)⫽sl,u,r,b . For the
where ␾
nodes on the boundary, there will be three contributions from
the three nearest neighbors; the fourth contribution will be
⫾i, depending on whether current is injected or extracted
from the node. Thus we get a total of N 2 coupled secondorder differential equations, where (p, ␾ ) form a total of 2N 2
variables to be updated at each step. We can visualize the
whole set of equations, in a matrix form, as
CṖ⫽RP¿ISÄD,
⌽̇ÄP,
current are the response of the array to a cyclic external
current. The I-V curve obtained this way is hysteretic for
␤ 0 ⬎1, as expected.
In this algorithm every step requires a large (N 2 ⫻N 2 )
matrix multiplication. This requires N 4 multiplications for
each update of the variables. Assuming that in each step we
change the current fed to the array by I C /N, where I C is the
average critical current of each bond, the total complexity of
the simulation increases as ⬃N 5 .
共4兲
where P, D, S, and ⌽ are column vectors of length N 2 each,
and C, R, and I are N 2 ⫻N 2 matrices. These matrices do not
change with time. However, C is a singular matrix, irrespective of whether the array is regular or disordered. This singularity implies that all the variables in the problem are not
independent. This introduces an extra complication in the
problem. In the mathematical literature such systems of
equations are called ‘‘differential algebraic equations’’
共DAE’s兲. They occur frequently in lattice-related problems.
Physically it is not difficult to trace the equation of constraint here. We have written equations for the phase of each
junction, whereas only the phase differences are of consequence. We can add an arbitrary number to each phase
ensuring
兺 ␾ i ⫽0.
all nodes
This is achieved by setting all the numbers in any row 共say,
the last兲 of the matrix C to be equal to 1 and the corresponding entry in the column vector D to be 0. The modified C is
no longer singular and can be inverted. The set of equations
is solved by using a fourth-order Runge-Kutta method with
variable time stepping.23
For each current we allow the system to evolve until ␶
⫽2500; the first 500 time units are regarded as stabilization
time and discarded. The voltage across the array is then averaged between 500⬍ ␶ ⬍2500. The value of the external
current is then increased by a small amount and the abovementioned cycle repeated, until the injected current reaches
the desired maximum value. After that it is decremented in
identical steps. The recorded values of the voltage vs drive
For ordered arrays the computation can be made much
faster, exploiting the special form the matrix C takes in such
a case. The matrix C is then a ‘‘connectivity matrix’’; the
(rជ ,rជ⬘ ) element of the matrix will be ⫺1 if the sites rជ and rជ⬘
are connected by a bond, and the diagonal element will give
the coordination number of the site. Himbergen et al.24 noted
that this particular form of the matrix allows the multiplication to be carried out in ⬃N ln N steps. Because the eigenvectors of this matrix are of the form exp(ikជ•rជ), the multiplication with the inverse of the matrix can be viewed as two
discrete Fourier transforms, amenable to ‘‘fast Fourier transform’’ techniques. The technique was improved by
Dominguez et al.25 and applied to several array geometries
soon after.26 Unfortunately, this fast algorithm requires that
the capacitance of all the bonds be same, even though disorder in R and I C can be handled. We however need to see the
effect of disorder in all the bond parameters. Consequently
we had to use straightforward matrix multiplication—i.e., the
‘‘slow’’ technique.
B. Critical currents of the films
We next investigate the temperature dependence of the
experimentally measured critical currents. Here we plot the
normalized critical current against the reduced temperature.
Figure 5 shows data from several Bi films, gathered from
different runs and on different substrates. All the points appear to collapse on a simple power law curve, given by
I c 共 T 兲 /I c 共 0 兲 ⫽1⫺ 共 T/T c 兲 4 .
共5兲
A similar behavior of Sn films is also shown in Fig. 5. The
critical current of a weak link is related to the superconducting gap (⌬) by the well-known Ambegaokar-Baratoff relation
I c R N ⫽ 共 ␲ ⌬ 共 T 兲 /2e 兲 tanh ⌬ 共 T 兲 / 共 2k B T 兲 .
共6兲
As T→T c , ⌬→0, we can expand the tanh term to show that
I c ⬃⌬ 2 . This suggests that for these films ⌬ vanishes as
冑1⫺(T/T c ) 4 . Near T c , we have 冑1⫺(T/T c ) 4
⬇2 冑1⫺(T/T c ), which is consistent with the behavior of a
BCS gap, as far as the leading power is concerned. However,
inserting the BCS result, 关i.e., ⌬(T)/⌬(0)⬇1.74冑1⫺T/T c ]
in the Ambegaokar-Baratoff relation leads to the well-known
prediction that near T c , the slope 关 I c R N /(T⫺T c ) 兴 should be
635 ␮ V/K. For all the Bi and Sn films studied by us, we
found this slope to be 960⫾20 ␮ V/K.
TABLE I. The estimates of the number of junctions for Bi films.
This estimate is for a 1 mm⫻1 mm area.
Data set
Thickness
T c 共K兲
R N (⍀)
N
4I c (0)R N /T c
No. 36
No. 25
53 Å Bi
65 Å Bi/Ge
65 Å Bi
85 Å Bi/Ge
4.32
4.5
4.29
5.23
421
236
252
135
942
651
730
473
953
954
955
954
No. 35
Omel’yanchuk form comprises recursive functions, no
simple expression exists for the dependence of the gap or
critical current on reduced temperature.28 However, the corrections for the presence of disorder to the AmbegaokarBaratoff relation, near T⫽0, are within a factor of 2 for both
the dirty and clean limits. This increase for the dirty limit
may partly explain the reason for the steeper dependence of i
on t observed in this work. Very close to T c it coincides with
Eq. 共6兲.
As an interesting aside, we mention that Dynes et al. have
shown in tunneling experiments on quenched Sn and Pb
films29 that the ratio 2⌬(0)/k B T c remains close to 3.5 共the
BCS value兲 for sheet resistances upto at least 4 k⍀. The
agreement was better for Sn than Pb; this may be expected as
Pb is a ‘‘strong-coupling’’ superconductor.
C. Estimate of the number of junctions in a film
FIG. 5. Critical currents of a set of Bi and Sn films of low sheet
resistance, in the regime E J ⰇE c . Both show a similar power-law
behavior over the entire temperature range.
It is interesting to note here that a similar behavior of
⌬(T) 关i.e., ⬃ 冑1⫺(T/T c ) 4 ] over the entire range of temperature has been observed in fabricated Josephson junction
arrays.27 In such fabricated arrays, one naturally expects the
spread of the junction parameters to be quite narrow, and in
this sense the disorder is considerably less than that for a
random array, such as the quench condensed films we study
in this work.
Though the data sets are restricted to T/T c ⬎0.3, in all the
experiments, the flattening of the curves allows us to make
an extrapolation of the critical current to the T→0 limit. This
is important, since I c (0) is very simply related to the superconducting gap by Eq. 共6兲. This relation can be refined further to account for the presence of disorder, as has been done
by Kulik and Omel’yanchuk.28 In fact, disordered films may
show a much better match to the Kulik-Omel’yanchuk form,
rather than the Ambegaokar-Baratoff form. Since the Kulik-
Using the model of the disordered array as the background and the extrapolated values of the critical current at
T⫽0, we show that an estimate of the number of junctions
in the film can be made. The estimate shows that not all grain
boundaries may be acting as junctions or weak links. It also
supports the possibility that the first layer of atoms that stick
to the substrate may have a significantly different
structure15,16 than the subsequent upper layers. In such cases
a slightly more uniform lower layer may offset the phasebreaking effect of a considerable number of grain boundaries.
If superconducting behavior of each grain follows approximately the BCS model, then we should have the zerotemperature gap ⌬(0)⫽1.76k B T c . Such an assumption is
certainly valid in the vicinity of T c , deviations from this
being important only at lower temperatures, as discussed
above. The average critical current of each junction is I c /N,
where I c and T c are experimentally measured. The number
of junctions acting in parallel should then be given by
I c R N /N⫽ ␲ ⌬ 共 0 兲 /2e⫽ ␲ 共 1.76k B T c 兲 /2e.
共7兲
The total number of junctions 共over 1 mm⫻1 mm) is then
approximately N 2 . Using the critical current data shown and
their T c we find the Bi films the parameters given in Table I.
We find that there are ⬃106 junctions/mm2 in a ⬇60 Å
film. As expected the number of junctions reduce when the
film thickness is increased. This is expected as many of the
gaps and voids between grains may be filling up as more
material is deposited, reducing the number of junctions.
Based on this we obtain the following values for ␻ , R, and C
for the 85 Å film at T⫽0:
␤ ⫽4.5, ␻ ⫽3.63⫻1012 rad/sec, I c /bond⫽8.9 ␮ A, C bond
⫽9⫻10⫺15 F,
which are reasonable values.
We find that at low temperatures I c tends to a constant
value and so does I r . This implies that the ratio I r /I c and
hence ␤ also varies very little at low temperatures, which is
to be expected. Near T c the behavior is dominated by the
variation of the critical current of the bond. The values of the
capacitance and resistance 共normal state兲 do not vary much
with temperature. We also find that for this value of thickness, the film is in a regime where the charging energy and
‘‘Coulomb blockade’’ are negligible compared to the
Josephson-coupling energy at low temperatures. The value of
the capacitance will, however, reduce drastically if the thickness of the film is lesser. In fact a recent optical frequency
measurement17 of the intergrain capacitance, in much thinner
Pb films, reported a value of ⬃2⫻10⫺19 F, which can be
compared with the values we have inferred above. In such
cases single-electron tunneling effects may be expected to
play a very important role in transport processes. It is interesting to compare the values we have estimated with typical
values of ‘‘fabricated’’ regular Josephson-junction arrays.
From the published literature we pick one work27 we have
already cited earlier. We find that the arrays used had a typical junction capacitance of 1 –3⫻10⫺15 F, junction area of
⬃1 ␮ m2 , and junction resistance of 4 –150 k⍀. These values, particularly those of the junction capacitance and ‘‘unit
cell’’ area that we have estimated, are of the same order.
However, the values of Josephson-coupling energy 共and
hence I c /bond兲 reported by them are much less compared to
deposited films. Thus the screening effects of the supercurrents flowing in the films may be expected to be much
greater than in fabricated arrays. The current density would
surely have a lot of spatial variation—if, however, we deliberately neglect this aspect and calculate a supercurrent density a film 共taking the nominal thickness to be the average
thickness兲 can support before going normal; for the 85 Å
film we get a number ⬃105 A/cm2 , which can be compared
with typical values reported for disordered films of the copper oxide superconductors.
D. Possible Kosterlitz-Thouless transition
in presence of strong disorder
In an ordered array of Josephson junctions, at finite temperatures vortex-antivortex pairs are generated spontaneously. These vortices may be visualized as a circulating pattern of the ‘‘phase variable’’ in neighboring islands, several
characteristics of these vortices have been studied in superfluids and arrays. One of the atomic scale disordered systems
that has been studied with respect to the Kosterlitz-Thouless
共KT兲 transition is the quenched Hg-Xe mixture.30 In this section we investigate whether some of the observed characteristics of the I-V curves of disordered Sn and Bi films can be
attributed to a Kosterlitz-Thouless-type transition.
FIG. 6. Data from a 40 Å film on bare quartz 共upper panel兲 and
with 10 Å Ge underlayer 共lower panel兲. Although the I-V are hysteretic, the hysteresis is not shown for purposes of clarity. In both
cases, a few I-V curves below T c have a linear part. Data taken at
lower temperatures do not show a power-law behavior 共linear region in a log-log plot兲.
Figure 6 shows a set of I-V curves taken at various temperatures. All the curves show a clear critical current and
retrapping current—and a transition to the normal-state resistance of 1210 and 942 ⍀ for this particular film. For a few
temperatures just below T c , the ‘‘superconducting state’’
shows dissipation. The resistance remains constant over three
to four decades of current and hence there is no self-heating
effect. It is tempting to identify the appearance of an Ohmic
dissipative state with the unbinding of vortices—the
‘‘Kosterlitz-Thouless’’ transition. However, we need to be
cautious with such an identification. One of the signatures of
the 70 Å Bi film data show this feature—see Fig. 7 共upper
panel兲—whereas for many other thicknesses the I-V curves
below the possible T KT do not have any region where a V
⬀I 3 dependence is obvious. Any such behavior should have
shown up clearly in a log-log plot. The 70 Å film also shows
another characteristic signature of KT transition 共lower
panel兲. Above T KT the resistance rises with temperature as
R 共 T 兲 ⫽R 0 exp关 ⫺ ␣ / 冑T⫺T KT 兴 .
The ‘‘best fit’’ is shown in Fig. 7. We have a mean-field T c
⫽3.55 K and T KT ⫽3.01 K for this film. With increasing
thickness, the resistive transition becomes steeper and the
range of temperature over which the KT-type behavior may
be seen also narrows down. For a 100 Å Bi film (R 䊐
⫽106 ⍀, data not shown兲 we found that the region narrows
down to less than 100 mK. This is to be expected since the
thicker films approach 3D behavior and the KT transition is
restricted to 2D systems. An important aspect that remains
unaddressed is the robustness of the KT transition to disorder. However, since vortices are macroscopic objects, which
average over large areas of the films—compared to microscopic atomic-scale disorder—the KT transition may be expected to remain quite robust in presence of disorder.
IV. CONCLUSION
FIG. 7. Data from a 70 Å Bi film shows power-law region in
logarithmic plot 共upper panel兲 as well as the predicted dependence
of R on T 共lower panel兲. Both of these features together identify a
KT transition. Hysteresis is not shown for the sake of clarity.
KT transition is that, below T KT there is a regime where the
voltage increases as the cube of the current. The reason for
this is that below T KT all the free vortices are generated by
the current itself. The current exerts oppositely directed
‘‘Lorentz force’’ on the vortex and the antivortex—and hence
tends to create free vortices by breaking the pairs. The number of vortices just below T KT increases as I 2 and hence the
voltage rises approximately as I 3 . Even though we may see
such linear regions in a logarithmic plot, we do not always
find this predicted cubic dependence in that region of temperature. Experiments on proximity coupled arrays31 have
shown both the linear regime in a logarithmic plot and cubic
dependence of voltage on current. In our case we find that
We have shown that several properties of quench condensed films of Sn and Bi can be understood by visualizing
them as strongly disordered arrays of Josephson junctions.
Superconductor-insulator transitions in such arrays are well
known and predicted to occur around E j ⬇E c . 32,33 Referring
to Eq. 共2兲 we can see that it should occur in the vicinity of
R N ⬇R Q but not necessarily exactly at R N ⫽R Q . A simple
power-law behavior of the critical current of these films
共with R 䊐 ⬍500 ⍀) is found, and the observed I-V characteristics indicate a possible vortex-unbinding transition in these
films. The picture of Josephson coupling between superconducting patches in the film implies that there is a strong
variation of carrier density in the film itself. This bears a
strong resemblance to the percolation-type description of
2DEG 共Ref. 21兲 that has been successfully used to describe
some aspects of the observed metal-insulator transition in Si
metal-oxide–semiconductor field-effect transistors 共MOSFET’s兲. To some extent computer simulations can reproduce
the behavior of the actual films. The difficulties in doing
numerical work on strongly disordered systems are well
known. Though a renormalization group analysis of the KT
transition has been done,30 the effect of strong screening 共i.e.,
large self-inductance兲 in a disordered array remains to be
investigated.
ACKNOWLEDGMENTS
This work was supported by DST and UGC, Government
of India. K.D.G. thanks CSIR, Government of India, for support. We acknowledge discussions with Professor A. M.
Goldman and Professor C. J. Adkins.
*Present address: Department of Condensed Matter Physics, Weiz-
mann Institute of Science, Rehovot, Israel.
†
Electronic address: chandra@physics.iisc.ernet.in
1
D.B. Haviland, Y. Liu, and A.M. Goldman, Phys. Rev. Lett. 62,
2180 共1989兲.
2
G. Sambandamurthy, K. Das Gupta, V.H.S. Murthy, and N. Chandrasekhar, Solid State Commun. 115, 427 共2000兲.
3
K. Das Gupta, G. Sambandamurthy, Swati S. Soman, and N.
Chandrasekhar, Phys. Rev. B 63, 104502 共2001兲.
4
E. Bielejec, J. Ruan, and Wenhao Wu, Phys. Rev. Lett. 87,
036801 共2001兲.
5
H.M. Jaeger, D.B. Haviland, B.G. Orr, and A.M. Goldman, Phys.
Rev. B 40, 182 共1989兲.
6
B.G. Orr, H.M. Jaeger, and A.M. Goldman, Phys. Rev. Lett. 53,
2046 共1984兲.
7
B.G. Orr, H.M. Jaeger, and A.M. Goldman, Phys. Rev. B 32, 7586
共1985兲.
8
N. Nishida, S. Okuma, and A. Asamitsu, Physica B 169, 478
共1991兲.
9
S.J. Lee and J.B. Ketterson, Phys. Rev. Lett. 64, 3078 共1990兲.
10
Y. Liu and A.M. Goldman, Mod. Phys. Lett. B 8, 277 共1994兲.
11
D. Ephron, A. Yazdani, A. Kapitulnik, and M.R. Beasely, Phys.
Rev. Lett. 76, 1529 共1996兲.
12
D. Das and S. Doniach, Phys. Rev. B 60, 1261 共1999兲.
13
P. Phillips and D. Dalidovich, Phys. Rev. B 65, 081101 共2002兲.
14
T.V. Ramakrishnan, Phys. Scr. T27, 24 共1989兲.
15
K.L. Ekinci and J.M. Valles, Jr., Phys. Rev. B 58, 7347 共1998兲.
16
K.L. Ekinci and J.M. Valles, Jr., Phys. Rev. Lett. 82, 1518 共1999兲.
17
P.F. Henning, C.C. Homes, S. Maslov, G.L. Carr, D.N. Basov, B.
Nikolić, and M. Strongin, Phys. Rev. Lett. 83, 4880 共1999兲.
18
A.V. Danilov, S.E. Kubatkin, I.L. Landau, and L. Rinderer, J.
Low Temp. Phys. 103, 35 共1996兲.
19
I.A. Parshin, I.L. Landau, and L. Rinderer, Phys. Rev. B 54, 1308
共1996兲.
20
I.L. Landau, I.A. Parshin, and L. Rinderer, J. Low Temp. Phys.
108, 305 共1997兲.
21
Y. Meir, Phys. Rev. Lett. 83, 3506 共1999兲.
22
M. Tinkham, Introduction to Superconductivity 共McGraw-Hill,
New York, 1996兲.
23
W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery,
Numerical Recipes in C 共Cambridge University Press, Cambridge, England, 1993兲, Chap. 16.
24
H. Eikmans and J.E. van Himbergen, Phys. Rev. B 41, 8927
共1990兲.
25
D. Dominguez, J.V. Jose, A. Karma, and C. Weiko, Phys. Rev.
Lett. 67, 2367 共1991兲.
26
S. Datta, S. Das, D. Sahadev, R. Mehrotra, and S. Shenoy, Phys.
Rev. B 54, 3545 共1996兲.
27
P. Delsing, C.D. Chen, D.B. Haviland, Y. Harada, and T. Cleason,
Phys. Rev. B 50, 3959 共1994兲.
28
I.O. Kulik and A.N. Omel’yanchuk, JETP Lett. 21, 96 共1976兲.
29
J.M. Valles, Jr., R.C. Dynes, and J.P. Garno, Phys. Rev. B 40,
6680 共1989兲.
30
A.M. Kadin, K. Epstein, and A.M. Goldman, Phys. Rev. B 27,
6691 共1983兲.
31
D.J. Resnick, J.C. Garland, J.T. Boyd, S. Shoemaker, and R.S.
Newrock, Phys. Rev. Lett. 47, 1542 共1981兲.
32
L.J. Geerligs, M. Peters, L.E.M. de Groot, A. Verbruggen, and
J.E. Mooij, Phys. Rev. Lett. 63, 326 共1989兲.
33
W. Zwerger, in Quantum Coherence in Mesoscopic Systems, edited by B. Kramer 共Plenum Press, New York, 1991兲.